Let \((\mathbf {a}, \, \mathbf {d})\) . By Lemma REF ,
there exists an open neighbourhood \(V\)  of \(\mathbf {a}\) 
in \(M^n\)  such that the function \(\zeta (\cdot , \, \mathbf {d})\colon V\rightarrow \mathbb {R}^{2\mathfrak {g}}\) 
is smooth. Let \(\xi \in \mathcal {L}(\mathbf {a}, \, \mathbf {d})\) .
We need to find a neighbourhood \(U\subseteq V\)  of \(\mathbf {a}\) 
and a smooth map \(\Xi \colon U\rightarrow \mathrm {Harm}^1(M)\) 
such that \(\Xi (\mathbf {a}) = \xi \)  and
\( \alpha \Xi (\mathbf {b}) + \zeta (\mathbf {b}, \, \mathbf {d}) \in (2\pi \mathbb {Z})^{2\mathfrak {g}}\qquad \textrm {for any } \mathbf {b}\in U\) 
Here \(\alpha \)  is the linear map defined by (REF ).
(Equation (REF ) is equivalent to \(\Xi (\mathbf {b})\in \mathcal {L}(\mathbf {b}, \, \mathbf {d})\) ,
due to (REF ).) As \((2\pi \mathbb {Z})^{2\mathfrak {g}}\)  is discrete,
(REF ) is equivalent to require
that \(\mathbf {b}\mapsto \alpha \Xi (\mathbf {b}) + \zeta (\mathbf {b}, \, \mathbf {d})\) 
be constant, that is
\( \alpha \Xi (\mathbf {b}) + \zeta (\mathbf {b}, \, \mathbf {d})= \alpha \xi + \zeta (\mathbf {a}, \, \mathbf {d})\qquad \textrm {for any } \mathbf {b}\in U\) 
Since \(\alpha \)  is invertible [1]},
the existence, uniqueness and regularity of a map \(\Xi \) 
satisfying (REF ) follow by the implicit function theorem.
